Decoherence-Free Entropic Gravity: Model and Experimental Tests

Alex J. Schimmoller,1, ∗ Gerard McCaul,1, † Hartmut Abele,2, ‡ and Denys I. Bondar1, §

1Tulane University, New Orleans, LA 70118, USA2Technische Universitat Wien, Atominstitut, Stadionallee 2, 1020 Wien, Austria

(Dated: July 23, 2021)

Erik Verlinde’s theory of entropic gravity [JHEP 2011, 29 (2011)], postulating that gravity is not afundamental force but rather emerges thermodynamically, has garnered much attention as a possibleresolution to the quantum gravity problem. Some have ruled this theory out on grounds that entropicforces are by nature noisy and entropic gravity would therefore display far more decoherence thanis observed in ultra-cold neutron experiments. We address this criticism by modeling linear gravityacting on small objects as an open quantum system. In the strong coupling limit, when the model’sunitless free parameter σ goes to infinity, the entropic master equation recovers conservative gravity.We show that the proposed master equation is fully compatible with the qBounce experiment forultra-cold neutrons as long as σ & 250 at 90% confidence. Furthermore, the entropic master equationpredicts energy increase and decoherence on long time scales and for large masses, phenomena whichtabletop experiments could test. In addition, comparing entropic gravity’s energy increase to thatof the Diosi-Penrose model for gravity induced decoherence indicates that the two theories areincompatible. These findings support the theory of entropic gravity, motivating future experimentaland theoretical research.

I. INTRODUCTION

The theory of entropic gravity challenges the assump-tion that gravity is a conservative force, i.e., one that isproportional to the gradient of a potential energy. En-tropic gravity instead postulates that gravity is an en-tropic force that points in the direction of maximum en-tropy [1].

The definition of entropic forces follows from the firstlaw of thermodynamics, δQ = dU + δW , which equatesheat supplied to a system δQ to the change in the sys-tem’s internal energy dU plus work done δW . If thereis a change in entropy dS = δQ/T with no change ininternal energy, then there is work done δW = TdS.The entropic force is the one performing the work F =δW/dx = TdS/dx due to the entropy gradient.

While Newtonian gravity is conservative, Verlinde’sproposal that gravity is entropic in nature [1] has gar-nered much attention. A simple argument in favor of thishypothesis goes as as follows: Bekenstein [2] argued thata particle of mass m held by a string just outside a blackhole will effectively be absorbed once the particle ap-proaches within one Compton wavelength, ∆x = ~/(mc),of the event horizon. Since the particle is so close to theevent horizon, it is unknown whether the particle still ex-ists or has been destroyed. So, the particle has gone frombeing in a pure “exists” state to either an “exists” or “de-stroyed” state with equal probabilities. Hence, the blackhole’s entropy has increased by ∆S = kb ln(2). Newton’ssecond law F = ma immediately follows from the en-tropic force definition F = T∆S/∆x after substituting i)

∗ aschimmo@tulane.edu† gmccaul@tulane.edu‡ hartmut.abele@tuwien.ac.at§ dbondar@tulane.edu

the amended form of Bekenstein’s formula ∆S = 2πkb,ii) the Compton wavelength ∆x, and iii) Unruh’s for-mula [3–5], kbT = ~a/(2πc), connecting acceleration withtemperature. Such a derivation of Newton’s second lawis valid for a black hole – an extreme concentration ofmass. Verlinde postulates this conclusion to be valid forall masses, which should be represented by holographicscreens [6].

Verlinde’s theory has undergone scrutiny, especiallyover the invocation of holographic screens and the Unruhformula [7–10], although these criticisms acknowledge aconnection between thermodynamics and gravity [11–13].Recently, an extension to non-holographic screens hasbeen established [14].

The aim of this work is to refute another prevailingcriticism of entropic gravity [7–10] that entropic forcesare by nature too noisy and thus destroy quantum co-herence. In particular, it has been argued in [10] that ifgravity were an entropic force, then it could be modeledas an environment in an open quantum system. Brow-nian motion is not observed for small masses inside theenvironment, so these small objects must be very stronglycoupled to the gravity environment. But the strong cou-pling must lead to ample wavefunction collapse and quan-tum decoherence. However, such decoherence is not ob-served in cold neutron experiments [15]. Thus, entropicgravity cannot be true according to [7, 8].

We disprove this argument by constructing (Sec. II)a non-relativistic model [Eq. (5)] for quantum particles(e.g., neutrons) interacting with gravity represented byan environment. According to this model, the strongerthe coupling to the reservoir, the lower the decoherence.Moreover, arbitrarily low decoherence can be achievedby simply increasing the positive dimensionless couplingconstant σ, which is a free parameter of this model. Inthe limit σ → ∞, the model recovers Newtonian grav-ity as a potential force [Eq. (3)]. A comparison of our

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model with data from the recent qBounce experiment[16] provides a lower bound σ & 500 (Secs. III and IV).We discuss some of entropic gravity’s physical impli-cations including monotonic energy increase and mass-dependent decoherence in Sec. V. A relationship to theDiosi-Penrose gravitational model is also discussed.

II. A MODEL OF ENTROPIC GRAVITYACTING NEAR EARTH’S SURFACE

In this section, we develop a near-Earth model of en-tropic gravity acting on quantum particles. Consider aparticle of mass m a small distance x above Earth’s sur-face in free-fall. In the classical case, the particle’s dy-namics are dictated by Newton’s equations of motion

d

dtx =

1

mp,

d

dtp = −mg (1)

where p is the particle’s momentum and g is the gravi-tational acceleration. In the quantum regime, however,these equations must be recast in the language of oper-ators and expectation values. This is accomplished viathe Ehrenfest theorems [17]

d

dt〈x〉 =

1

m〈p〉 , d

dt〈p〉 = −mg. (2)

Free fall of a quantum particle, whose state is repre-sented by the density matrix ρ, in a linear gravitationalpotential is described by the Liouville equation [18]

dρ

dt= − i

~

[p2

2m+mgx, ρ

]. (3)

Recalling that the expectation value for an observableO is given by 〈O〉 = Tr(Oρ), it can easily be shownthat Eq. (3) satisfies the free-fall Ehrenfest theorems (2).Equation (3) is the conservative model for free-fall. Thepurity of a quantum state ρ is given by Tr(ρ2). The pu-rity reaches its maximum value of unity if and only if thedensity matrix corresponds to the sate representable by awave function. It is an important feature of Eq. (3) thatit preserves the purity, i.e., Eq. (3) maintains coherence.

Equation (3) is not the only one capturing free-falldynamics (2). In fact, within the language of openquantum systems [19], there are an infinite number ofmaster equations which satisfy the above Ehrenfest the-orems [20]. It has been shown in [20] that for arbitraryG(p) and F (x), the Ehrenfest theorems

d

dt〈x〉 = 〈G(p)〉 , d

dt〈p〉 = 〈F (x)〉 (4)

can be satisfied by coupling a closed system with theusual Hamiltonian H = p2/(2m) + U(x) to a series oftailored environments. We take advantage of this factto model gravity as an environment in an open quantumsystem fashion [19].

In the simplest case, a linear gravitational potentialcan be treated as a single dissipative environment and thefree-fall dynamics (2) are satisfied by the master equationof the Lindblad form [20]

dρ

dt= − i

~

[p2

2m, ρ

]+D(ρ), (5)

D(ρ) =mgx0σ

~

{exp

(− ix

x0σ

)ρ exp

(+ix

x0σ

)− ρ},

(6)

where

x0 =

(~2

2m2g

)1/3

(7)

is a characteristic length and σ is a unitless, positive cou-pling constant, which is a free parameter in the model[21]. Note that the Hamiltonian in Eq. (5) only containsthe kinetic energy term, and the linear gravitational po-tential is replaced by the dissipator (6). We propose touse Eq. (5) as the model for entropic gravity acting onquantum particles near Earth’s surface.

To elucidate how the dissipator (6) mimics a lineargravitational potential, we employ the Hausdorff expan-sion with the assumption σ →∞ to obtain

dρ

dt=− i

~

[p2

2m+mgx, ρ

]+

mg

x0~σ

(xρx− 1

2x2ρ− 1

2ρx2)

+O

(1

σ2

). (8)

Thus, utilizing large values of the coupling constant σ,the master equation for entropic gravity (5) can approx-imate the conservative equation (3) with an arbitrarilyhigh precision.

The argument put forth in Refs. [7, 8] against entropicgravity has the following fault: It is based on the as-sumption that the evolution of a neutron’s initial purestate to a mixed one is generated by a non-Hermitiantranslation operator (see Eq. (12) of [7]) leading to theSchrodinger equation with a non-Hermitian Hamiltonian(see Eq. (16) of [7]). While non-Hermitian correctionsto the Schrodinger equation have been historically usedto incorporate some aspects of dissipation, such an ap-proach suffers from physical inconsistencies [22] and hasbeen abandoned in the modern theory of open quantumsystems. Hence, instead of Eq. (12) from Ref. [7] thatreads

ρ(z + ∆z) ≡ U ρ(z)U†, U U† = 1, (9)

the Kraus representation (see, e.g, Ref. [19]) for the evo-lution ρ(z)→ ρ(z + ∆z) should have been used

ρ(z + ∆z) ≡∑n

Knρ(z)K†n,∑n

K†nKn = 1. (10)

The Kraus representation furnishes the most generaldescription for evolution of open quantum systems. The

3

only requirement used to arrive at Eq. (10) is that themapping ρ(z)→ ρ(z+∆z) should be completely positive.The latter is a stronger requirement than the fact thatphysical evolution preserves the positivity of a densitymatrix. Finally, we note that a Lindblad master equation[such as, e.g., Eq. (5)] can be recast in a Kraus form.

If the O(σ−2

)term is dropped in Eq. (8), then the

resulting Eq. (8) describes a particle undergoing a con-tinuous quantum measurement of its position [19, 23].The entropic master equation (5) interprets gravity as acontinuous measurement process extracting informationabout the position of a massive particle. The extractionof information is responsible for the entropy creation [24].As a result, the purity of the quantum system is no longerpreserved.

The rate of change of the purity induced by evolutiongoverned by Eq. (5) is estimated as σ →∞,

d

dtTr(ρ2) = −2

mg

x0~σTr(ρ2x2 − (ρx)2

)+O

(1

σ2

).

(11)

It is shown in [20] that Tr(ρ2x2 − (ρx)2

)≥ 0; thus,

the purity is monotonically decreasing. Furthermore, thelarger the σ, the more purity is preserved. Since we canelect to make σ arbitrarily large in our model, the origi-nal criticism of entropic gravity not maintaining quantumcoherence can no longer be considered valid.

The proposed entropic master equation (5) obeys avariant of the equivalence principle (see, e.g., Refs. [25,26]). According to [27], the strong equivalence principlestates that “all test fundamental physics is not affected,locally, by the presence of a gravitational field.” Hence,dynamics induced by a homogeneous gravitational fieldmust be translationally invariant. Equation (5) is knownto be translationally invariant [28–33].

Since Verlinde’s theory treats gravity as a thermody-namically emergent force, it is not appropriate to quan-tize gravity and talk about the existence of gravitons[34, 35]. However, our entropic master equation (5) phe-nomelogically hints at gravitons. Equations similar toEq. (5) have long been employed for the nonperturbativedescription of a quantum system undergoing collisionswith a background gas of atoms or photons [31–33, 36–38]. Transferring this microscopic picture, the dissipa-tor (6) can be interpreted as describing colissions of amassive quantum particle with a bath of gravitons; more-over, ~/(x0σ) stands for the momentum of a graviton. Topreserve purity σ must be large, which makes the momen-tum of a graviton infinitesimally small. This conclusionis compatible with the fact that detecting a graviton re-mains a tremendous challenge [39], which might becomefeasible [40].

A plethora of models for gravitation induced decoher-ence, which describe quantum matter interacting with astochastic gravitational background, has been put forth[35]. It is worth pointing out that some of these mod-els mathematically resemble the entropic master equa-tion (5); in particular, the models of time fluctuations

[41–43], spontaneous collapse [44–47], and the Diosi-Penrose model [48–53]. However, despite mathematicalresemblance, they can make very different predictionsfrom Eq. (5) (see Sec. V). We also note that Lindblad-like master equations have been recently emerged in post-quantum classical gravity [54, 55], where a quantum sys-tem interacts with classical space-time.

III. MODELING THE QBOUNCEEXPERIMENT

Now that the free-fall model for entropic gravity hasbeen established [Eq. (5)], it is desirable to see how itcompares to results of the qBounce experiment [16].This experiment was performed at the beam position forultra-cold neutron at the European neutron source at theInstitut Laue-Langevin in Grenoble and uses gravity res-onance spectroscopy [56] to induce transitions betweenquantum states of a neutron in the gravity potential ofthe earth. In region I of this experiment, neutrons areprepared in a known mixture of the first three quantumbouncer energy states (see Appendix A). These neutronsthen traverse a 30 cm horizontal boundary which oscil-lates with variable frequency ω and oscillation amplitudea, inducing Rabi oscillations between the “bouncing-ball”states of neutrons. In Figs. 2 and 3 below, the oscillationstrength is defined as aω. Finally in region III, neutronspass through a state selector, leaving neutrons in an un-known mixture of the three lowest energy states to becounted. To model this experiment, the free-fall masterequation (5) must be amended to account for the oscillat-ing boundary, and simulations must account for variableneutron times-of-flight and the unknown selection of neu-trons in region III.

The simplest way to model the boundary is by mod-ifying the Ehrenfest theorems. For a system with thegeneral Hamiltonian

H = p2/(2m) + U(x), (12)

and the boundary condition 〈x = 0|ψ〉 = 0, the secondEhrenfest theorem reads

d

dt〈p〉 = 〈−U ′(x)〉+

~2

2m

(d

dx〈x|ψ〉

)∣∣∣∣x=0

(d

dx〈ψ|x〉

)∣∣∣∣x=0

= 〈−U ′(x)〉+~2

4m〈δ′′(x)〉 , (13)

where δ(x) is the Dirac delta function, defined as∫ ∞−∞

dxδ(n)(x− x′)f(x) = (−1)(n)f (n)(x′). (14)

Thus modifying the Hamiltonian H to include the bound-ary term,

H =p2

2m+ U(x)− ~2

4mδ′(x). (15)

4

recovers the desired Ehrenfest theorem (13).In order to make the boundary oscillate, one simply

needs to add a sinusoidal term inside of the Dirac deltafunction:

H =p2

2m+ U(x)− ~2

4mδ′(x− a sin(ωt)). (16)

Here, a is the oscillation amplitude.In the particular case of potential gravity [Eq. (3)], a

neutron’s dynamics while inside the qBounce apparatusis described by the Liouville equation:

dρ

dt= − i

~

[p2

2m+mgx− ~2

4mδ′ (x− a sin(ωt)) , ρ

].

(17)

Meanwhile, the entropic case [Eq. (5)] gives

dρ

dt= − i

~

[p2

2m− ~2

4mδ′ (x− a sin(ωt)) , ρ

]+ D(ρ). (18)

Here, the kinetic and boundary terms are inside the com-mutator and D(ρ) is the gravity environment (6). Be-cause D(ρ) is translationally invariant, the oscillatingboundary does not alter the dissipator (6).

For simulations of the qBounce experiment, we trans-form the equations of motion into the reference frame ofthe oscillating boundary (see Appendix B). After apply-ing the change of variables x = x − a sin(ωt) and trans-lating x → x, the conservative model’s Liouville equa-tion (17) becomes [57]

dρ

dt= − i

~

[p2

2m+mgx− ~2

4mδ′(x)− aω cos(ωt)p, ρ

],

(19)

and the entropic Lindblad equation (18) reads

dρ

dt= − i

~

[p2

2m− ~2

4mδ′(x)− aω cos(ωt)p, ρ

]+D(ρ).

(20)

Differentiating with respect to the unitless time τ =tmgx0/~ yields the unitless conservative Liouville equa-tion

dρ

dτ= −i

[h+ ξ + w, ρ

], (21)

along with the unitless entropic Lindblad equation

dρ

dτ= −i

[h+ w, ρ

]+ σ

(DρD† − ρ

). (22)

Here, h represents the kinetic energy and boundary

terms, ξ gives the potential energy term, w accounts forthe accelerating frame and D gives the first exponentialinside the D(ρ) term. Matrix elements for these opera-tors are given in Appendix C. Equations (21) and (22)are used in the following simulations.

Now that proper master equations have been estab-lished for region II, how long must they run? The time-of-flight tf for each neutron is determined by its horizontalvelocity v = 0.30(mgx0)/(~τf ), ultimately determiningfinal state populations Pj(τf ) = Tr (ρ(τf )|Ej〉〈Ej |). Inthis experiment, neutrons are measured to have horizon-tal velocities v between 5.6 and 9.5 m/s. We elect tomake the horizontal neutron velocity v an additional freeparameter in the model confined to this range. While thischoice in modeling does not capture the range of veloc-ities contributing to the overall transmission, results inSec. IV indicate that this assumption does not diminishthe overall point of the paper.

Finally, a full model of the qBounce experiment [16]requires modeling the state selection in region III. Thestate selector consists of an upper mirror positioned justabove the attainable height of a ground state neutron.However, higher states leak into the detector as well. Wethus define relative transmission (neutron count rate withthe oscillating boundary divided by the count rate with-out oscillation) to be a linear combination of the threelowest energy state populations:

T = c0P0 + c1P1 + c2P2, (23)

where c0, c1, and c2 are unknown, positive coefficientsto be determined from experimental data as explained inAppendix D. Since the state selector is designed to scat-ter away excited neutrons, the physical and engineeringconsideration leads to the constraint c0 ≥ c1 ≥ c2.

IV. SIMULATING THE QBOUNCEEXPERIMENT

With the results of Sec. III, we can effectively sim-ulate the qBounce experiment [16]. In region I of theexperiment, neutrons are prepared initially as an incoher-ent mixture with 59.7% population in the ground state,34.0% in the first excited state, 6.3% in the second excitedstate and no population in higher states. Thus, the ini-tial state of simulated neutrons is the incoherent mixtureρ(0) = 0.597|E0〉〈E0|+0.340|E1〉〈E1|+0.063|E2〉〈E2|. Inregion II, neutrons interact with gravity and the oscillat-ing boundary. The density matrix evolves according toeither the conservative (21) or entropic (22) unitless mas-ter equations, with frequency ω and oscillation strengthaω determined by the experimental setup. After the in-teraction time τf (determined by the free velocity pa-rameter v), simulated neutrons have effectively passedthrough region II of the experiment. We calculate thefinal populations P0, P1, and P2.

We perform minimization of χ2 over the space of thefive parameters: c0, c1, c2, v, and σ (see Appendix D fordetails). An agreement between the theory and experi-ment can be observed in Figs. 1, 2, and 3. As Eq. (8)predicts, transmission values for entropic simulations ap-proach those of the conservative model as σ increases.This is to say, conservative gravity can be recovered with

5

large enough σ in the entropic model, and decoherence ef-fects are therefore unnoticed. In particular, a good agree-ment of the experimental data with the entropic modelis observed when σ equals 500. Furthermore, χ2 analysisshown in Fig. 4 reveals that simulations with σ & 250 fitthe experimental data with 90% confidence. Note thatat σ = 500 the values of χ2 for conservative and entropicgravity coincide. In conclusion, we take 500 to be thelower bound for σ.

In total, the entropic model of the qBounce experi-ment consists of five free parameters: σ, v, c0, c1, andc2. For entropic simulations with σ . 250, the best-fitvelocity hovers around the lower limit of 5.6 m/s. Asσ → ∞, the best-fit velocity approaches 6.58 m/s. Thetransmission coefficients c0, c1, and c2 equal to 1.46, 0.50and 0.50, respectively, for σ = 500, and approach 1.28,0.55, and 0.55, respectively, as σ →∞.

0 10 20 30 40Data Point Index

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Rela

tive

Tran

smiss

ion

= 100= 500= 103

Conservative GravityQbounce Experiment

FIG. 1. Comparing the qBounce experiment [16] with pre-dictions of the master equation for entropic gravity [Eq. (18)]as well as the conservative gravity [Eq. (17)]. All data pointsfrom the experiment are visible with corresponding frequencyω and oscillation strength aω data replaced with a single in-dex on the horizontal axis. 20 states are accounted for innumerical propagation of Eqs. (17) and (18).

V. DISCUSSION AND FUTURE DIRECTIONS

We have shown that a linear gravitational potentialcan be modeled by an environment coupled to neutrons.This entropic gravity model overcomes the criticism putforth in Ref. [10] since the master equation (5) is ca-pable of maintaining both strong coupling and negligibledecoherence and is fully compatible with the qBounceexperiment [16]. Moreover, the entropic model recoversthe conservative gravity (3) as σ →∞. Our findings pro-vide support for the entropic gravity hypothesis, whichmay spur further experimental and theoretical inquires.

Let us compare the predictions of the entropic masterequation (5) and the Diosi-Penrose (D-P) model [35, 58].

Consider the total energy operator H = p2/(2m) +mgx.

While the expected total energy 〈H〉 remains constant inthe conservative case (3), the entropic model’s rate of the

2500 3000 3500 4000 4500Oscillation Frequency (Hz)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Rela

tive

Tran

smiss

ion

= 100= 500= 103

Conservative GravityQbounce Experiment

FIG. 2. Comparing the qBounce experiment [16] with pre-dictions of the master equation for entropic gravity [Eq. (18)]as well as the conservative gravity [Eq. (17)] by varying os-cillation frequency (ω) when the oscillation strength (aω) isset to 2.05 mm/s. 20 states are accounted for in numericalpropagation of Eqs. (17) and (18). σ is a free parameter inthe entropic gravity master equation. When σ & 500 theexperiment agrees well with entropic gravity.

0 1 2 3 4 5Oscillation Strength (mm/s)

0.2

0.4

0.6

0.8

1.0

1.2

Rela

tive

Tran

smiss

ion

= 100= 500= 103

Conservative GravityQbounce Experiment

FIG. 3. Comparing the qBounce experiment [16] with pre-dictions of the master equation for entropic gravity [Eq. (18)]as well as the conservative gravity [Eq. (17)] by varying oscilla-tion strength (aω) with the oscillation frequency (ω) set to thethe transition between the ground and third excited states ofthe “bouncing ball” problem [ω = ω03 = (E3 − E0)/~ = 4.07kHz]. σ is a free parameter in the entropic gravity masterequation. When σ & 500 the experiment agrees well withentropic gravity.

expected energy change is given by

d

dt

⟨H⟩

=g~

2x0σ. (24)

That is, under entropic gravity, the test particle’s totalenergy increases at a rate ∝ 1/σ regardless of the ini-tial state. Hence, the entropic model avoids a thermalcatastrophe in the large coupling limit (σ → ∞), un-like the D-P model. According to the latter, the rate ofenergy increase (given by Eq. (94) in Ref. [35]) equalsmG~/(4

√πR3

0), where G is the gravitational constantand R0 is a coarse-graining parameter set to the nu-cleon’s radius, 10−15 m. For a neutron, the D-P modelpredicts the rate of energy increase to be 1.66×10−27 W(= 10.4 neV/s), while the entropic model prediction is

6

200 400 600 800 100050

55

60

65

70

752

Entropic GravityConservative Gravity

2 = 2min + 2.7

FIG. 4. χ2 as a function of σ. The gray area representsthe 90% confidence interval. χ2

min is the minimum χ2 valueamong the simulated results. When σ & 250, entropic gravityfalls within this region. When σ & 500, entropic gravity fitsexperimental data as well as conservative gravity.

significantly lower: 1.76 × 10−31 W (= 1.1 peV/s) as-suming σ = 500 (see Sec. IV). For the entropic modelto display as much energy increase as the D-P modelpredicts, σ would need to be 0.05, much less than whatis permitted by the qBounce experiment as shown inSec. IV. Moreover, for a 1 kg mass, the D-P model pre-dicts a rate of energy increase ≈ 1 Watt! Such a sig-nificant quantity should be readily noticeable. Compar-atively, the entropic model predicts the rate of energyincrease of only 0.125 pW when σ = 500. Raising R0 cansignificantly reduce the D-P model’s energy increase, butthere is no physical justification for larger values of R0.We also note that recent extensions to the D-P modelto include the gravitational backreaction [59] suffer fromthe same issue. As shown in Appendix F, the additionalterms arising from the inclusion of a semiclassical fieldserve only to double the rate of energy increase. Com-paratively, there is no known upper bound on σ, andenergy increase vanishes as σ →∞.

We believe that the lower bound σ = 500, deduced inSec. IV from the qBounce experiment, is highly likelyto be an underestimation. A more realistic lower boundshould be σ & 4.6 × 105. Let us describe how the lattervalue could be confirmed experimentally. According toEq. (24), a neutron will gain energy ∆E within a time∆t,

∆t =2x0σ

g~∆E. (25)

Assume the neutron is initially prepared in the groundstate |E0〉 of the “bouncing ball”. Then, we let itevolve for the time approaching the neutron’s lifetime∆t = 881.5 s and measure the final state. If it jumped tothe first excited state |E1〉, then according to Eq. (25),the neutron must have gained ∆E ≥ E1 − E0 implyingthat σ ≤ 4.6 × 105. If the neutron does not reach |E1〉,then σ > 4.6 × 105. Storage experiments with neutronsmight provide these limits [57].

The entropic master equation (5) predicts gravity in-duced decoherence albeit at a much lower rate than, e.g.,

the D-P model. In Appendix E, we show that if td isthe decoherence time for a particle of mass m, then thedecoherence time t′d for mass M is t′d = (M/m)−1/3td.Hence, the larger the mass, the faster the decoherence.Moreover, measuring the decoherence times would alsodirectly identify σ. The recent experiment [60] that ob-served optomechanical nonclassical correlations involvinga nanopartcile could perform such a test.

Although the proposed entropic gravity model is lim-ited to the low-energy, near-Earth regime, its physical im-plications provide a glimpse into several open cosmolog-ical questions. As Ref. [35] mentions regarding collapsegravitational models, entropic gravity’s non-unitarity dy-namics could resolve the black hole information paradox[61, 62], and its runaway energy (24) could pose solu-tions to the dark energy [63], cosmological inflation, andquantum measurement problems [64]. With greater re-striction of σ from precision experiments and better un-derstanding of its physical implications at all time andenergy scales, entropic gravity can be further explored asa feasible gravitational theory.

ACKNOWLEDGMENTS

H.A. and D.I.B. are grateful to Wolfgang Schleichand Marlan Scully for inviting us to the PQE-2019 con-ference, where this collaboration was conceived. H.A.thanks T. Jenke for fruitful discussions. A.J.S. andD.I.B. wish to acknowledge the Tulane Honors SummerResearch Program for funding this project. G.M. andD.I.B. are supported by the Army Research Office (ARO)(grant W911NF-19-1-0377), Defense Advanced ResearchProjects Agency (DARPA) (grant D19AP00043), andAir Force Office of Scientific Research (AFOSR) (grantFA9550-16-1-0254). The views and conclusions containedin this document are those of the authors and shouldnot be interpreted as representing the official policies,either expressed or implied, of ARO, DARPA, AFOSR,or the U.S. Government. The U.S. Government is au-thorized to reproduce and distribute reprints for Govern-ment purposes notwithstanding any copyright notationherein. H.A. gratefully acknowledges support from theAustrian Fonds zur Forderung der WissenschaftlichenForschung (FWF) under contract no. P 33279-N.

Appendix A: Solving the Schrodinger Equation Fora Bouncing Ball

In this section, we solve the quantum bouncing ballproblem (as is done in [65]). Consider the time-independent Schrodinger equation for a particle of massm experiencing a linear gravitational potential U(x) =mgx and an infinite potential barrier at x = 0. We wishto find the the eigenvalues E and eigenvectors |E〉 such

7

that

Hc |E〉 = E |E〉 , where (A1)

Hc =p2

2m+mgx. (A2)

Applying 〈x| to equation (A1), the equation can berewritten as(

d2

dx2− 2m

~2[mgx− E]

)〈x|E〉 = 0, (A3)

and the infinite potential barrier manifests itself in theboundary condition

〈x = 0|E〉 = 0. (A4)

It is easy to confirm that the solutions to equation (A3)are given by

〈x|E〉 = c1Ai

(ξ − E

mgx0

)+ c2Bi

(ξ − E

mgx0

), (A5)

where

x0 =

(~2

2m2g

)1/3

, (A6)

ξ = x/x0, (A7)

and c1, c2 are constants, and Ai and Bi are the twolinearly-independent solutions to the Airy equation(

d2

dy2− y)w(y) = 0, w = Ai(y), Bi(y). (A8)

Considering the normalization condition∫ ∞0

| 〈x|E〉 |2dx = 1, (A9)

we exclude Bi since Bi(x) → ∞ as x → ∞ [66]. Apply-ing (A9) to (A3) with c2 = 0, we get our normalizationcoefficient:

c = c(E) =

[x0

∫ ∞0

dξAi2(ξ − E

mggx0

)]−1/2. (A10)

Thus, solutions in the coordinate representation are givenby

〈x|E〉 =Ai(ξ − E

mgx0

)[x0∫∞0dξAi2

(ξ − E

mgx0

)]1/2 . (A11)

Applying the boundary condition (A4) yields eigen-values En = −mgx0an+1, where n = 0, 1, 2, ... and ajdenotes the jth zero of Ai. By convention, the energyeigenstates of a system are numbered beginning with zeroto signify the ground state, whereas the zeroes of a func-tion are numbered beginning with one, hence the nth

energy state corresponding to the (n + 1)th zero of theAiry function. Corresponding eigenfunctions are givenby

〈x|En〉 =Ai(ξ + an+1)[

x0∫∞0dξAi2 (ξ + an+1)

]1/2 . (A12)

The set of eigenvectors {|En〉}∞n=0 forms an orthonormalbasis.

Appendix B: The Quantum Bouncer with anOscillating Boundary: Change of Variables

In this section, the Schrodinger equation used to modelthe qBounce experiment is converted to the referenceframe of the oscillating boundary. The following treat-ment closely follows Ref. [57]. Consider the 1D time-dependent Schrodinger equation for a particle with po-tential energy U(x), along with an infinite potential bar-rier, which oscillates with a frequency ω and amplitudea about the point x = 0:

i~d

dt|ψ(t)〉 = H |ψ(t)〉 (B1)

where

H =p2

2m+ U(x)− ~2

4mδ′ (x− a sin(ωt)) . (B2)

When 〈x| is applied on the left to both sides of (B1), onegets the Schrodinger equation in the coordinate represen-tation:

i~d

dt〈x|ψ(t)〉 =

{− ~2

2m

∂2

∂x2+ U(x)

− ~2

4mδ′ (x− a sin(ωt))

}〈x|ψ(t)〉 . (B3)

Given the infinite potential barrier, one can impose theboundary condition

〈x = a sin(ωt)|ψ(t)〉 = 0. (B4)

The goal is now to convert (B3) to the reference frameof the oscillating mirror. Given the change of variablesx = x− a sin(ωt), it is easy to show that

∂2

∂x2=

∂2

∂x2, (B5)

d

dt=

∂

∂t+

(∂x

∂t

)∂

∂x

=∂

∂t− aω cos(ωt)

∂

∂x. (B6)

Thus, the equation of motion (B3) in the reference frameof the oscillating barrier becomes

i~∂

∂t〈x|ψ(t)〉 = {H0 +W (x, t)} 〈x|ψ(t)〉, (B7)

8

where

H0 = − ~2

2m

∂2

∂x2+ U(x)− ~2

4mδ′(x), (B8)

W (x, t) = U(a sinωt) + i~aω cos(ωt)∂

∂x, (B9)

U(ˆx+ a sinωt) = U(ˆx) + U(a sinωt), (B10)

〈x|ψ(t)〉 = 〈x|ψ(t)〉 . (B11)

Notice how when the time-dependent term W (x, t) = 0,the equation of motion reduces to the time-independentquantum bouncing ball problem of Appendix A. Tosimplify notation, return x → x and ψ(t) → ψ(t)in Eqs. (B7)-(B9) and rewrite the original Schrodingerequation (B7) from the mirror’s reference frame:

i~∂

∂t|ψ(t)〉 =

{H0 + W

}|ψ(t)〉 , (B12)

H0 =p2

2m+ U(x)− ~2

4mδ′(x), (B13)

W = U(a sinωt)− aω cos(ωt)p. (B14)

Note that the p operator in this last equation is onlypresent so as to be transformed into −i~ ∂

∂x when 〈x| isreapplied. When we convert our Schrodinger equation(B12) into the density matrix formalism, we get that

dρ

dt= − i

~

[p2

2m+ U(x)− ~2

4mδ′(x)− aω cos(ωt)p, ρ

].

(B15)

Furthermore, consider the D(ρ) operator in the en-tropic model given by equation (6). Under the changeof variables x = x−a sin(ωt), D(ρ) is invariant under thechange of variables since

exp

(− i(

ˆx+ a sinωt)

x0σ

)ρ exp

(+i(ˆx+ a sinωt)

x0σ

)

= exp

(− iˆx

x0σ

)ρ exp

(+iˆx

x0σ

). (B16)

Appendix C: qBounce Simulation Matrix Elements

In order to simulate the qBounce experiment usingQuTiP [67], the master equations (19) and (20) mustfirst be made unitless. This can be accomplished by dif-ferentiating with respect to unitless time τ = (tmgx0)/~.The conservative master equation becomes

dρ

dτ= − i

mgx0

[p2

2m+mgx− ~2

4mδ′(x)− aω cos(ωt)p, ρ

](C1)

and the entropic Lindblad equation becomes

dρ

dτ= − i

mgx0

[p2

2m− ~2

4mδ′(x)− aω cos(ωt)p, ρ

]+D(ρ)

mgx0.

(C2)

Sandwiching these master equations between 〈Ej | on theleft and |Ek〉 [see Eq. (A12)] on the right yields the unit-less conservative master equation

dρ

dτ= −i

[h+ ξ + w, ρ

], (C3)

along with the unitless entropic master equation

dρ

dτ= −i

[h+ w, ρ

]+ σ

(DρD† − ρ

), (C4)

where

hjk = −aj+1δjk −∫∞0dξξAi(ξ + aj+1)Ai(ξ + ak+1)

NjNk,

(C5)

ξjk =

∫∞0dξξAi(ξ + aj+1)Ai(ξ + ak+1)

NjNk, (C6)

wjk = +i

(4m

~g

)1/3

(aω) cos(ωt)∫∞0dξAi (ξ + aj+1) d

dξAi (ξ + ak+1)

NjNk, (C7)

Djk =

∫∞0dξ exp(−iξ/σ)Ai(ξ + aj+1)Ai(ξ + ak+1)

NjNk,

(C8)

Nj =

[∫ ∞0

dξAi2(ξ + aj+1)

]1/2. (C9)

Here, h gives the boundary and kinetic energy term. ξ =x/x0 is the unitless position operator, w accounts for

the accelerating frame, D is the first exponential term inD(ρ), and Nj is the normalization factor. In a similarfashion, we can show that the matrix elements for theposition and momentum operators x and p, along withδ′′(x) in the |Ei〉 basis are given by

xjk = x0

∫∞0dξξAi(ξ + aj+1)Ai(ξ + ak+1)

NjNk, (C10)

pjk = − i~x0

∫∞0dξAi(ξ + aj+1) ddξAi(ξ + ak+1)

NjNk,

(C11)

δ′′jk(ξ) =

[ddξAi (ξ + aj+1)

]ξ=0

[ddξAi (ξ + ak+1)

]ξ=0

NjNk.

(C12)

In all our numerical simulations we use 20× 20 matrices.

Appendix D: χ2 minimization

To simulate region III of the qBounce experiment [16]using the entropic model (22), for each value of the neu-tron’s velocity 5.6 m/s ≤ v ≤ 9.5 m/s and each value

9

of the coupling constant 102 ≤ σ ≤ 103, we solve thefollowing convex optimization problems

minimizec0, c1, c2

χ2(σ, v)

subject to c0 ≥ c1 ≥ c2 ≥ 0,(D1)

where χ2(v) is the chi-square goodness of fit

χ2(σ, v) =∑a,ω

[Texp(a, ω)− Ttheor(a, ω;σ, v)]2

εexp(a, ω)2, (D2)

Texp(a, ω) is experimentally measured relative trans-mission with corresponding error εexp(a, ω), andTtheor(a, ω; v) is the theoretical transmission [Eq. (23)]

Ttheor(a, ω;σ, v) =

2∑j=0

cjPj

(a, ω;σ, τf = 0.30

mgx0~v

).

(D3)

Here, Pj(a, ω;σ, τf ) are the final population of statej = 0, 1, 2 as a function of the driving frequency andstrength. The summation in Eq. (D2) is done over mea-sured data. We find c0, c1, and c2 in Eq. (D1) using theoptimizer CVXPY [68, 69]. As a convex optimizationtask, the problem (D1) has a unique solution. Note thatthe optimal solution (c0, c1, c2) depends on v and σ. InFigs. 1, 2, and 3, we compare (D3) with the experimen-tal measurements Texp(a, ω) by fixing the velocity v suchthat it minimizes χ2(σ, v) [Eq. (D2)] for a given value ofσ.

Appendix E: Entropic Gravity Mass Dependence

Let us answer the question: How does the entropicmaster equation (5) change when a different mass is intro-duced, say M = κm? Substituting m→ κm in Eq. (18)(explicitly and and also implicitly in x0) gives

dρ

dt=− i

~

[p2

2mκ− ~2

4mκδ′(x), ρ

]+mgx0σκ

1/3

~

{e−

ixκ2/3

x0σ ρe+ixκ2/3

x0σ − ρ}. (E1)

However, eigenfunctions (A12) are also non-trivially de-pendent on mass. In particular, eigenfunctions for ob-jects of mass M are given by

〈x|E′n〉 (M) =Ai(ξκ2/3 + an+1

)κ1/3[

x0∫∞0dξAi2 (ξ + an+1)

]1/2 , (E2)

where again ξ = x/x0 and x0 =(

~2

2m2g

)1/3. Solving

for matrix elements of D using the above eigenfunction

definition yields

Djk = 〈E′j |D|E′k〉 (E3)

=

∫dx exp

(− ixκ

2/3

x0σ

)⟨E′j |x

⟩〈x|E′k〉 (E4)

=

∫dξκ2/3e−

iξκ2/3

σ Ai(ξκ2/3 + aj+1)Ai(ξκ2/3 + ak+1)

NjNk.

(E5)

Scaling the integration by ξκ2/3 → ξ will thus yield theoriginal matrix elements (C8). In a similar fashion, ma-trix elements for the Hamiltonian (C5) are recovered.Hence, the master equation becomes

dρ

dt= − imgx0κ

1/3

~

[h, ρ]

+mgx0σκ

1/3

~

(DρD† − ρ

),

(E6)

with matrix elements given by (C5) and (C8), exactlythe same as with the original mass. Differentiating withrespect to τM = mgx0κ

1/3/~ gives

dρ

dτM= −i

[h, ρ]

+ σ(DρD† − ρ

), (E7)

whose right hand side is equal to that of master equation(C4) (wihout the oscillation term w) in which mass isequal to m.

Consider the purity rate of change with respect to τM :

d

dτMTr(ρ2) = −2σTr

(ρ2 − ρDρD†

). (E8)

Employing the Hausdorff expansion with respect to σ toDρD† gives

d

dτMTr(ρ2) = − 2

σTr(ρ2ξ2 − (ρξ)2

)+O

(1

σ2

), (E9)

where ξ = x/x0.Thus, purity decay for different masses follows the

same form. Only time scale is changed. If td is the timescale for mass m, then the time scale t′d for mass M is

t′d = κ−1/3td, where κ = M/m.As an illustration, let us select M to be the Planck

mass and m – the neutron’s mass, then κ = 1.30× 1019.Say some effect is observed for the neutron during itslifetime of about 881.5 s. Then the same effect is theo-retically observable for the Planck mass, but at a time of375 µs. Likewise, say the Planck mass experiences somepurity decay within one second of interacting with thegravity environment. Then to observe the same puritydecay in the neutron, it would take 2.35× 106 s (≈ 27.2days) far beyond the neutron’s lifetime.

Appendix F: Spontaneous Localisation Models

One of the principal efforts to combine classical gravi-tational fields with quantum dynamics are spontaneous

10

localisation models. Such models introduce additionalnon-linear and stochastic terms to quantum dynamics,so as to guarantee the spatial localisation of matter atmacroscopic scales while leaving the microscopic dynam-ics unchanged [35]. This is achieved by specifying thatthe additional collapse operators correspond to the localmass density m(x) =

∑imiδ

(3)(x − xi), coupled to astochastic process. For a Markovian noise, the dynamicsof the stochastically averaged density matrix ρ are givenby [35]:

d

dtρ = −i

[H, ρ

]−1

4

∫d3x

∫d3y K(x−y) [m(x), [m(y), ρ]] ,

(F1)where K(x − y) is the kernel of the stochastic processand ~ ≡ 1.

The connection of such spontaneous collapse modelsto gravity was made explicit in the Diosi-Penrose (D-P)model [35, 58], where the stochastic kernel was chosento be the Newtonian gravitational potential, K(x) = G

|x| .

While such a choice of kernel provides a natural connec-tion to gravitation, it also necessitates a coarse-grainingprocedure, without which the integral in the second termwill diverge. This is achieved by convolving the massdensity with a Gaussian of width R0:

fR(x) =(2πR2

0

)−3/2 ∫d3y exp

(−|x− y|2

2R20

)f(y).

(F2)In order to obtain a finite dissipator, it is critical that themass density operator is coarse-grained in this manner,i.e. m(x)→ mR(x)

In more recent work, this model has been extended toinclude the backreaction of the quantised matter on thegravitational field [59]. Introducing the Newtonian field

operator

Φ(x) = −G∫

d3ym(y)

|x− y|, (F3)

leads to an extra term in Eq.(F1) of the form:

− 1

16πG

∫d3x

[∇ΦR(x),

[∇ΦR(x), ρ

]]. (F4)

This gravitational backreaction is not only local, but ofprecisely the same form as the collapse term in Eq. (F1).This is most easily seen in the Fourier representation foreach term. Using m(k) = F [m(x)] we obtain:

1

16πG

∫d3x

[∇ΦR(x),

[∇ΦR(x), ρ

]]=G

8π2

∫d3k

exp(−R2

0|k|2)

|k|2[m(k),

[m†(k), ρ

]]=

1

4

∫d3x

∫d3y K(x− y) [m(x), [m(y), ρ]] . (F5)

Consequently, the addition of the gravitational back-reaction term in Eq.(F4) amounts to a doubling of thestrength of the decoherence term in Eq.(F1) [59]. Thishas important consequences when considering the ener-getic implications of these models. Without a dissipativeterm, energy is conserved in neither model, and it is easyto show [35] that for a single particle of mass m in the D-

P model, the rate of change of energy is dEDP (t)dt = mG~

4√πR3

0

[35]. When including the backreaction, this rate is simplydoubled.

In both cases, the rate of energetic change is stronglydetermined by the chosen coarse-graining cut-off R0. Anatural approach to choosing this cut-off is to argue thatit should correspond to the Compton wavelength of anucleon, R0 ≈ 10−15 m. As shown in the main texthowever, such a value leads to enormous rates of change,and therefore predicts a thermal catastrophe.

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